Download Test of Mathematics for University Admission Test Specification

Test of Mathematics for University Admission
Specification for November 2016 Examination
Structure of the Test
The test will consist of two one-hour papers, taken one after the other.
Each paper will consist of 20 multiple-choice questions.
Questions across the two papers carry equal weight and there will be no penalty for incorrect
answers, so candidates are advised to attempt all questions.
There is no formulae booklet for this test; students are expected to understand and recall all
relevant formulae.
Candidates may not use calculators.
The details of the papers are as follows:
Paper 1: Mathematical Knowledge and Application
Time: 1 hour
Content: 20 multiple-choice questions
Requirements: Section 1 below
This paper will test the candidate’s ability to apply their mathematical knowledge in a
variety of contexts. Candidates will be expected to know and use the mathematical
content set out in Section 1 below.
Paper 2: Advanced Mathematical Thinking
Time: 1 hour
Content: 20 multiple-choice questions
Requirements: Sections 1 and 2 below
This paper will test the candidate’s ability to apply their conceptual knowledge to
constructing and analysing mathematical arguments. For this paper candidates are
expected to be familiar with the contents of Sections 1 and 2 below.
©UCLES 2016
2
SECTION 1
This section sets out the mathematical knowledge requirement for both papers of the test.
The content of Part 1 is almost all covered within the pure mathematics specification of an
AS level in mathematics, and the content of Part 2 is almost all covered within a Higher
Level GCSE mathematics course.
There is some duplication of content across Parts 1 and 2.
Candidates are advised to read through these specifications carefully to ensure they are
aware of all topics and areas that might be covered in the test.
Part 1
Algebra and functions
AF1
Laws of indices for all rational exponents.
AF2
Use and manipulation of surds; simplifying expressions that contain surds, including
rationalising the denominator; for example, simplifying
√5
3+2√5
, and
3
√7−2√3
.
AF3
Quadratic functions and their graphs; the discriminant of a quadratic function;
completing the square; solution of quadratic equations.
AF4
Simultaneous equations: analytical solution by substitution, e.g. of one linear and one
quadratic equation.
AF5
Solution of linear and quadratic inequalities.
AF6
Algebraic manipulation of polynomials, including:

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AF7
Expanding brackets and collecting like terms;
Factorisation and simple algebraic division (by a linear polynomial,
including those of the form 𝑎𝑎 + 𝑏, and by quadratics, including those
of the form 𝑎𝑥 2 + 𝑏𝑏 + 𝑐);
Use of the Factor Theorem and the Remainder Theorem.
Qualitative understanding that a function is a many-to-one (or sometimes just a oneto-one) mapping. Familiarity with the properties of common functions, including
𝑓(𝑥) = √𝑥 (which always means the ‘positive square root’) and 𝑓(𝑥) = |𝑥|
Sequences and series
SE1
SE2
SE3
SE4
Sequences, including those given by a formula for the 𝑛th term and those generated
by a simple recurrence relation of the form 𝑥𝑛+1 = 𝑓(𝑥𝑛 ).
Arithmetic series, including the formula for the sum of the first 𝑛 natural numbers.
The sum of a finite geometric series; the sum to infinity of a convergent geometric
series, including the use of |𝑟| < 1.
Binomial expansion of (1 + 𝑥)𝑛 for positive integer 𝑛, and for expressions of the form
𝑛
�𝑎 + 𝑓(𝑥)� for positive integer 𝑛 and simple 𝑓(𝑥); the notations 𝑛! and �𝑛𝑟�.
©UCLES 2016
3
Coordinate geometry in the (𝒙, 𝒚) plane
CG1
Equation of a straight line, including 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) and 𝑎𝑎 + 𝑏𝑏 + 𝑐 = 0;
conditions for two straight lines to be parallel or perpendicular to each other; finding
equations of straight lines given information in various forms.
CG2
Coordinate geometry of the circle: using the equation of a circle in the forms
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2 , and 𝑥 2 + 𝑦 2 + 𝑐𝑐 + 𝑑𝑑 + 𝑒 = 0.
CG3 Use of the following circle properties:
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The perpendicular from the centre to a chord bisects the chord;
The tangent at any point on a circle is perpendicular to the radius at that
point;
The angle subtended by an arc at the centre of a circle is twice the angle
subtended by the arc at any point on the circumference;
The angle in a semicircle is a right angle;
Angles in the same segment are equal;
The opposite angles in a cyclic quadrilateral add to 180°;
The angle between the tangent and chord at the point of contact is equal
to the angle in the alternate segment.
Trigonometry
1
TR1
The sine and cosine rules, and the area of a triangle in the form 𝑎𝑎 sin 𝐶.
2
The sine rule includes an understanding of the ‘ambiguous’ case (angle-side-side).
Problems might be set in 2- or 3-dimensions.
TR2
Radian measure, including use for arc length and area of sector and segment.
TR3
The values of sine, cosine, and tangent for the angles 0°, 30°, 45°, 60°, 90°.
TR4
TR5
TR6
The sine, cosine, and tangent functions; their graphs, symmetries, and periodicity.
Knowledge and use of tan 𝜃 =
sin 𝜃
cos 𝜃
and sin2 𝜃 + cos2 𝜃 = 1.
Solution of simple trigonometric equations in a given interval (this may involve the
1
use of the identities in TR5); for example: tan 𝑥 = − 3 for – 𝜋 < 𝑥 < 𝜋;
2
sin �2𝑥 +
©UCLES 2016
𝜋
�
3
1
2
2
√
= for −2𝜋 < 𝑥 < 2𝜋; 12 cos 𝑥 + 6 sin 𝑥 − 10 = 2 for 0° < 𝑥 < 360°.
4
Exponentials and Logarithms
EL1
EL2
𝑦 = 𝑎 𝑥 and its graph, for simple positive values of 𝑎.
Laws of logarithms:
𝑎𝑏 = 𝑐 ⟺ 𝑏 = log 𝑎 𝑐
log 𝑎 𝑥 + log 𝑎 𝑦 = log 𝑎 (𝑥𝑥)
𝑥
log 𝑎 𝑥 − log 𝑎 𝑦 = log 𝑎 � �
𝑦
including the special cases:
𝑘 log 𝑎 𝑥 = log 𝑎 �𝑥 𝑘 �
log 𝑎
1
= − log 𝑎 𝑥
𝑥
log 𝑎 𝑎 = 1
Questions requiring knowledge of the change of base formula will not be set.
EL3
The solution of equations of the form 𝑎 𝑥 = 𝑏, and equations which can be reduced to
this form, including those that need prior algebraic manipulation; for example,
32𝑥 = 4 and 25𝑥 − 3 × 5𝑥 + 2 = 0.
Differentiation
DF1
The derivative of 𝑓(𝑥) as the gradient of the tangent to the graph 𝑦 = 𝑓(𝑥) at a point.
In addition:

Interpretation of a derivative as a rate of change;

Second-order derivatives;

Knowledge of notation:
𝑑𝑑 𝑑 2 𝑦
,
𝑑𝑑 𝑑𝑑 2
Differentiation from first principles is excluded.
DF2
Differentiation of 𝑥 𝑛 for rational 𝑛, and related sums and differences. This might
require some simplification before differentiating; for example, the ability to
differentiate an expression such as
DF3
, 𝑓 ′ (𝑥), and 𝑓 ′′ (𝑥).
(3𝑥+2)2
1
𝑥2
could be required.
Applications of differentiation to gradients, tangents, normals, stationary points
(maxima and minima only), increasing [ 𝑓 ′ (𝑥) ≥ 0 ] and decreasing [ 𝑓 ′ (𝑥) ≤ 0 ]
functions. Points of inflexion will not be examined, although students are expected to
have a qualitative understanding of points of inflexion in the curves of simple
polynomial functions.
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Integration
IN1
IN2
IN3
Definite integration as related to the ‘area between a curve and an axis.’ Candidates
are expected to understand the difference between finding a definite integral and
finding the area between a curve and an axis. Integrals could be given with respect to
𝑥 or with respect to 𝑦.
Finding definite and indefinite integrals of 𝑥 𝑛 for 𝑛 rational, 𝑛 ≠ −1, and related sums
and differences, including expressions which require simplification prior to
integrating; for example, ∫(𝑥 + 2)2 𝑑𝑑, and ∫
1
𝑥2
𝑑𝑑.
An understanding of the Fundamental Theorem of Calculus and its significance to
integration. Simple examples of its use may be required in the two forms,
𝑏
IN4
(3𝑥−5)2
𝑑
𝑥
∫𝑎 𝑓(𝑥) 𝑑𝑑 = 𝐹(𝑏) − 𝐹(𝑎), where 𝐹 ′ (𝑥) = 𝑓(𝑥), and 𝑑𝑑 ∫𝑎 𝑓(𝑡) 𝑑𝑑 = 𝑓(𝑥).
Combining integrals with either equal or contiguous ranges;
5
5
5
for example, ∫2 𝑓(𝑥) 𝑑𝑑 + ∫2 𝑔(𝑥) 𝑑𝑑 = ∫2 [𝑓(𝑥) + 𝑔(𝑥)] 𝑑𝑑,
4
3
3
and ∫2 𝑓(𝑥) 𝑑𝑑 + ∫4 𝑓(𝑥) 𝑑𝑑 = ∫2 𝑓(𝑥) 𝑑𝑑.
IN5
Approximation of the area under a curve using the trapezium rule; determination of
whether this constitutes an overestimate or an underestimate.
IN6
Solving differential equations of the form
Graphs of Functions
𝑑𝑑
𝑑𝑑
= 𝑓(𝑥).
GF1
Recognise and be able to sketch the graphs of common functions that appear in this
specification: these include lines, quadratics, cubics, trigonometric functions,
logarithmic functions, exponential functions, square roots, and the modulus function.
GF2
Knowledge of the effect of simple transformations on the graph of 𝑦 = 𝑓(𝑥) as
represented by 𝑦 = 𝑎𝑎(𝑥), 𝑦 = 𝑓(𝑥) + 𝑎, 𝑦 = 𝑓(𝑥 + 𝑎), 𝑦 = 𝑓(𝑎𝑎), with the value of
𝑎 positive or negative. Compositions of these transformations.
GF3
GF4
Understand how altering the values of 𝑚 and 𝑐 affects the graph of 𝑦 = 𝑚𝑚 + 𝑐.
Understand how altering the values of 𝑎, 𝑏 and 𝑐 in 𝑦 = 𝑎(𝑥 + 𝑏)2 + 𝑐 affects the
corresponding graph.
GF5
Use differentiation to help determine the shape of the graph of a given function; this
might include finding stationary points (excluding inflexions) as well as finding when
the function is increasing or decreasing.
GF6
Use algebraic techniques to determine where the graph of a function intersects the
coordinate axes; appreciate the possible numbers of real roots a general polynomial
can possess.
GF7
Geometric interpretation of algebraic solutions of equations; relationship between the
intersections of two graphs and the solutions of the corresponding simultaneous
equations.
©UCLES 2016
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Part 2
Number
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Order, add, subtract, multiply and divide whole numbers, integers, fractions, decimals,
and numbers in index form.
Use the concepts and vocabulary of factor, multiple, common factor, highest common
factor (hcf), least common multiple (lcm), composite (i.e. not prime), prime number, and
prime factor decomposition.
Use the terms square, positive and negative square root, cube and cube root.
Use index laws to simplify, multiply, and divide integer, fractional, and negative powers.
Interpret, order, and calculate with numbers written in standard index form.
Understand equivalent fractions.
Convert between fractions, decimals, and percentages.
Understand and use percentage including repeated proportional change and
calculating the original amount after a percentage change.
Understand and use direct and indirect proportion.
Use ratio notation including dividing a quantity in a given ratio, and solve related
problems (using the unitary method).
Understand and use number operations, including inverse operations and the hierarchy
of operations.
Use surds and π in exact calculations; simplify expressions that contain surds, including
rationalising the denominator.
Calculate upper and lower bounds to contextual problems.
Approximate to a specified and appropriate degree of accuracy, including rounding to a
given number of decimal places or significant figures.
Know and use approximation methods to produce estimations of calculations.
©UCLES 2016
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Algebra
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Distinguish between the different roles played by letter symbols.
Manipulate algebraic expressions by collecting like terms; by multiplying a single term
over a bracket; by expanding the product of two linear expressions.
Use index laws in algebra for multiplication and division of integer, fractional, and
negative powers.
𝑐
Know and use of �𝑎𝑏 � = 𝑎𝑏𝑏
Set up and solve linear equations, including simultaneous equations in two unknowns.
Factorise quadratics, including the difference of two squares.
Simplify rational expressions by cancelling or factorising.
Set up quadratic equations and solve them by factorising.
Set up and use equations to solve problems involving direct and indirect proportion.
Derive a formula, substitute into a formula.
Change the subject of a formula.
Solve linear inequalities in one or two variables.
Generate terms of a sequence using term-to-term and position-to-term definitions.
Use linear expressions to describe the 𝑛th term of a sequence.
Use Cartesian coordinates in all 4 quadrants.
Recognise the equations of straight lines; understand 𝑦 = 𝑚𝑚 + 𝑐 and the gradients of
parallel and perpendicular lines.
Understand that the intersection of graphs can be interpreted as giving the solutions to
simultaneous equations.
Solve simultaneous equations, where one is linear and one is quadratic.
Recognise and interpret graphs of quadratic functions, simple cubic functions, the
reciprocal function, trigonometric functions and the exponential function 𝑦 = 𝑘 𝑥 for
simple positive values of 𝑘.
Construct linear functions from real-life problems; interpret graphs modelling real
situations.
©UCLES 2016
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Geometry
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Recall and use properties of angles at a point, on a straight line, perpendicular lines
and opposite angles at a vertex.
Understand and use the angle properties of parallel lines, intersecting lines, triangles
and quadrilaterals.
Calculate and use the sums of the interior and exterior angles of polygons.
Recall the properties and definitions of special types of quadrilateral.
Recognise and use reflectional and rotational symmetry of 2-dimensional shapes.
Understand congruence and similarity.
Use Pythagoras’ theorem in 2-dimensions and 3-dimensions.
Understand and construct geometrical proofs, including using circle theorems:
 The perpendicular from the centre to a chord bisects the chord;
 The tangent at any point on a circle is perpendicular to the radius at that
point;
 The angle subtended by an arc at the centre of a circle is twice the angle
subtended at any point on the circumference;
 The angle in a semicircle is a right-angle;
 Angles in the same segment are equal;
 The opposite angles in a cyclic quadrilateral add to 180°;
 The angle between the tangent and chord at the point of contact is equal
to the angle in the alternate segment.
Use 2-dimensional representations of 3-dimensional shapes.
Describe and transform 2-dimensional shapes using single or combined rotations,
reflections, translations, or enlargements, including the use of vector notation.
Understand and be able to use the standard trigonometric ratios: sin, cos, and tan.
Measures
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Calculate perimeters and areas of shapes made from triangles, rectangles and other
shapes.
Find circumferences and areas of circles, including arcs, segments and sectors.
Calculate the volumes and surface areas of right prisms, pyramids, spheres, cylinders,
cones and solids made from cubes and cuboids (formulae will be given for the sphere
and cone).
Use vectors, including the sum of two vectors, algebraically and graphically.
Use and interpret maps and scale drawings.
Understand and use the effect of enlargement for perimeter, area, and volume of
shapes and solids.
Convert measurements from one unit to another, including between imperial and metric
(conversion factors will be given for imperial/metric conversions).
Knowledge of the SI prefixes milli (m), centi (c), deci (d), and kilo (k) when used in
connection with any SI unit.
Recognise the inaccuracy of measurement.
Understand and use three-figure bearings.
Understand and use compound measures.
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Statistics
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Identify possible sources of bias.
Identify flaws in data collection sheets and questionnaires in an experiment or a survey.
Group, and understand, discrete and continuous data.
Extract data from lists and tables.
Design and use two-way tables.
Interpret bar charts, pie charts, grouped frequency diagrams, line graphs, and
frequency polygons.
Interpret cumulative frequency tables and graphs, box plots, and histograms (including
unequal class width).
Calculate and interpret mean, median, mode, modal class, range, and inter-quartile
range, including the estimated mean of grouped data.
Calculate average rates when combining samples or events, including solving problems
involving average rate calculations (e.g. average survival rates in different wards of
different sizes, average speed of a car over a journey where it has travelled at different
speeds).
Interpret scatter diagrams and recognise correlation; using lines of best fit. (The
calculation of regression lines is not required.)
Compare sets of data by using statistical measures or by interpreting graphical
representations of their distributions.
Probability
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Understand and use the vocabulary of probability and the probability scale.
Understand and use estimates or measures of probability, including relative frequency
and theoretical models.
List all the outcomes for single and combined events.
Identify different mutually exclusive outcomes and know that the sum of the
probabilities of all these outcomes is 1.
Construct and use Venn diagrams to solve union and intersection categorisation
problems and determine probabilities when required. Familiarity with the meaning and
use of the terms ‘union’, ‘intersection’, and ‘complement’ is required. The mathematical
notation for these (𝐴 ∪ 𝐵, 𝐴 ∩ 𝐵, and 𝐴′ or 𝐴𝑐 ) will not be required.
Know when to add or multiply two probabilities.
Understand the use of tree diagrams to represent outcomes of combined events:
 when the probabilities are independent of the previous outcome;
 when the probabilities are dependent on the previous outcome.
Compare experimental and theoretical probabilities.
Understand that if an experiment is repeated, the outcome may be different.
©UCLES 2016
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SECTION 2
This section sets out the scope of Paper 2. Paper 2 tests the candidate’s ability to think
mathematically: the paper will focus on testing the candidate’s ability to understand, and
construct, mathematical arguments in a variety of contexts. It will draw on the mathematical
knowledge outlined in SECTION 1 above.
The Logic of Arguments
Arg1 Understand and be able to use mathematical logic in simple situations:



The terms true and false;
The terms and, or (meaning inclusive or), not;
Statements of the form:
if A then B
A if B
A only if B
A if and only if B

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The converse of a statement;
The contrapositive of a statement;
The relationship between the truth of a statement and its converse and its
contrapositive.
Note: candidates will not be expected to recognise or use symbolic notation for any of these
terms, nor will they be expected to complete formal truth tables.
Arg2 Understand and use the terms necessary and sufficient.
Arg3 Understand and use the terms for all, for some (meaning for at least one), and
there exists.
Arg4 Be able to negate statements that use any of the above terms.
Mathematical Proof
Prf1
Follow a proof of the following types, and in simple cases know how to construct
such a proof:

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

Direct deductive proof (‘Since A, therefore B, therefore C, …, therefore Z,
which is what we wanted to prove.’);
Proof by cases (for example, by considering even and odd cases separately);
Proof by contradiction;
Disproof by counterexample.
Prf2
Deduce implications from given statements.
Prf3
Make conjectures based on small cases, and then justify these conjectures.
Prf4
Rearrange a sequence of statements into the correct order to give a proof for a
statement.
Prf5
Problems requiring a sophisticated chain of reasoning to solve.
©UCLES 2016
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Identifying Errors in Proofs
Err1
Identifying errors in purported proofs.
Err2
Be aware of common mathematical errors in purported proofs; for example, claiming
‘if 𝑎𝑎 = 𝑎𝑎, then 𝑏 = 𝑐’ or assuming ‘if sin 𝐴 = sin 𝐵, then 𝐴 = 𝐵’ neither of which are
valid deductions.
©UCLES 2016
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